LMFDB - Embedded newform 81.8.c.d.55.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,8,Mod(28,81)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$25.3031870642$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 17x^{2} + 16x + 256 $ x^4 - x^3 + 17x^2 + 16x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.2
Root			$-1.76556 + 3.05805i$ of defining polynomial
Character	$\chi$	$=$	81.55
Dual form			81.8.c.d.28.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(3.79669 - 6.57607i) q^{2} +(35.1702 + 60.9166i) q^{4} +(-32.9066 - 56.9959i) q^{5} +(369.202 - 639.477i) q^{7} +1506.08 q^{8} -499.745 q^{10} +(-2480.63 + 4296.58i) q^{11} +(-2983.62 - 5167.78i) q^{13} +(-2803.50 - 4855.80i) q^{14} +(1216.32 - 2106.72i) q^{16} +36651.6 q^{17} +22378.9 q^{19} +(2314.67 - 4009.12i) q^{20} +(18836.4 + 32625.6i) q^{22} +(25736.8 + 44577.4i) q^{23} +(36896.8 - 63907.1i) q^{25} -45311.5 q^{26} +51939.7 q^{28} +(34247.9 - 59319.0i) q^{29} +(-75327.4 - 130471. i) q^{31} +(87152.9 + 150953. i) q^{32} +(139155. - 241024. i) q^{34} -48596.8 q^{35} +489027. q^{37} +(84966.0 - 147165. i) q^{38} +(-49559.9 - 85840.2i) q^{40} +(-295318. - 511505. i) q^{41} +(421321. - 729750. i) q^{43} -348978. q^{44} +390859. q^{46} +(-613184. + 1.06207e6i) q^{47} +(139151. + 241016. i) q^{49} +(-280172. - 485272. i) q^{50} +(209869. - 363504. i) q^{52} +958904. q^{53} +326517. q^{55} +(556047. - 963101. i) q^{56} +(-260057. - 450432. i) q^{58} +(158135. + 273897. i) q^{59} +(14861.5 - 25740.8i) q^{61} -1.14398e6 q^{62} +1.63495e6 q^{64} +(-196362. + 340108. i) q^{65} +(-146512. - 253767. i) q^{67} +(1.28905e6 + 2.23270e6i) q^{68} +(-184507. + 319576. i) q^{70} +714537. q^{71} -3.96273e6 q^{73} +(1.85669e6 - 3.21587e6i) q^{74} +(787073. + 1.36325e6i) q^{76} +(1.83171e6 + 3.17262e6i) q^{77} +(-1.26902e6 + 2.19801e6i) q^{79} -160100. q^{80} -4.48492e6 q^{82} +(-831556. + 1.44030e6i) q^{83} +(-1.20608e6 - 2.08899e6i) q^{85} +(-3.19926e6 - 5.54128e6i) q^{86} +(-3.73602e6 + 6.47098e6i) q^{88} +4.64819e6 q^{89} -4.40624e6 q^{91} +(-1.81034e6 + 3.13560e6i) q^{92} +(4.65614e6 + 8.06467e6i) q^{94} +(-736415. - 1.27551e6i) q^{95} +(-7.35048e6 + 1.27314e7i) q^{97} +2.11325e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 9 q^{2} - 77 q^{4} - 180 q^{5} - 700 q^{7} + 3654 q^{8} + 2790 q^{10} - 10890 q^{11} + 5480 q^{13} - 29475 q^{14} + 15967 q^{16} + 32832 q^{17} + 32048 q^{19} - 12195 q^{20} - 60705 q^{22} - 24372 q^{23}+ \cdots + 45559476 q^{98}+O(q^{100}) $ 4 * q - 9 * q^2 - 77 * q^4 - 180 * q^5 - 700 * q^7 + 3654 * q^8 + 2790 * q^10 - 10890 * q^11 + 5480 * q^13 - 29475 * q^14 + 15967 * q^16 + 32832 * q^17 + 32048 * q^19 - 12195 * q^20 - 60705 * q^22 - 24372 * q^23 + 138880 * q^25 - 470520 * q^26 + 527750 * q^28 + 143280 * q^29 + 38708 * q^31 + 439965 * q^32 + 614088 * q^34 + 231300 * q^35 + 911240 * q^37 + 275382 * q^38 - 135765 * q^40 - 731880 * q^41 + 1088840 * q^43 + 1049130 * q^44 + 3298788 * q^46 - 1561500 * q^47 - 967164 * q^49 + 519660 * q^50 + 2106380 * q^52 + 5220936 * q^53 + 2007000 * q^55 + 650475 * q^56 + 720810 * q^58 - 1731960 * q^59 + 620192 * q^61 - 8572302 * q^62 - 2081678 * q^64 + 914400 * q^65 - 346600 * q^67 + 5559624 * q^68 - 3094425 * q^70 - 8484480 * q^71 - 6290380 * q^73 + 4267710 * q^74 + 2510486 * q^76 - 4864500 * q^77 - 10110616 * q^79 - 3411090 * q^80 - 4282380 * q^82 - 644202 * q^83 - 101520 * q^85 - 2313270 * q^86 - 9374715 * q^88 + 12042000 * q^89 - 41744000 * q^91 - 14795802 * q^92 + 3751290 * q^94 - 747180 * q^95 - 4098670 * q^97 + 45559476 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	3.79669	−	6.57607i	0.335583	−	0.581248i	−0.648013	−	0.761629i	$-0.724400\pi$
$2$	3.79669	−	6.57607i	0.335583	−	0.581248i	0.983597	+	0.180381i	$0.0577332\pi$
$3$		0			0
$3$		0			0
$4$	35.1702	+	60.9166i	0.274767	+	0.475911i
$5$	−32.9066	−	56.9959i	−0.117730	−	0.203915i	0.801138	−	0.598480i	$-0.204228\pi$
$5$	−32.9066	−	56.9959i	−0.117730	−	0.203915i	−0.918868	+	0.394565i	$0.870895\pi$
$6$		0			0
$7$	369.202	−	639.477i	0.406838	−	0.704664i	−0.587696	−	0.809082i	$-0.699965\pi$
$7$	369.202	−	639.477i	0.406838	−	0.704664i	0.994533	+	0.104418i	$0.0332981\pi$
$8$	1506.08			1.04000
$9$		0			0
$10$	−499.745			−0.158033
$11$	−2480.63	+	4296.58i	−0.561937	+	0.973304i	0.435390	+	0.900242i	$0.356611\pi$
$11$	−2480.63	+	4296.58i	−0.561937	+	0.973304i	−0.997327	+	0.0730623i	$0.976723\pi$
$12$		0			0
$13$	−2983.62	−	5167.78i	−0.376653	−	0.652382i	0.613920	−	0.789368i	$-0.289592\pi$
$13$	−2983.62	−	5167.78i	−0.376653	−	0.652382i	−0.990573	+	0.136986i	$0.956258\pi$
$14$	−2803.50	−	4855.80i	−0.273056	−	0.472947i
$15$		0			0
$16$	1216.32	−	2106.72i	0.0742381	−	0.128584i
$17$	36651.6			1.80935			0.904674	−	0.426104i	$-0.140114\pi$
$17$	36651.6			1.80935			0.904674	+	0.426104i	$0.140114\pi$
$18$		0			0
$19$	22378.9			0.748518			0.374259	−	0.927324i	$-0.377897\pi$
$19$	22378.9			0.748518			0.374259	+	0.927324i	$0.377897\pi$
$20$	2314.67	−	4009.12i	0.0646969	−	0.112058i
$21$		0			0
$22$	18836.4	+	32625.6i	0.377154	+	0.653250i
$23$	25736.8	+	44577.4i	0.441069	+	0.763955i	0.997769	−	0.0667591i	$-0.0212659\pi$
$23$	25736.8	+	44577.4i	0.441069	+	0.763955i	−0.556700	+	0.830714i	$0.687933\pi$
$24$		0			0
$25$	36896.8	−	63907.1i	0.472279	−	0.818012i
$26$	−45311.5			−0.505594
$27$		0			0
$28$	51939.7			0.447143
$29$	34247.9	−	59319.0i	0.260760	−	0.451649i	−0.705684	−	0.708526i	$-0.749360\pi$
$29$	34247.9	−	59319.0i	0.260760	−	0.451649i	0.966444	+	0.256877i	$0.0826936\pi$
$30$		0			0
$31$	−75327.4	−	130471.i	−0.454137	−	0.786589i	0.544501	−	0.838760i	$-0.316719\pi$
$31$	−75327.4	−	130471.i	−0.454137	−	0.786589i	−0.998638	+	0.0521715i	$0.983386\pi$
$32$	87152.9	+	150953.i	0.470172	+	0.814362i
$33$		0			0
$34$	139155.	−	241024.i	0.607187	−	1.05168i
$35$	−48596.8			−0.191589
$36$		0			0
$37$	489027.			1.58718			0.793591	−	0.608451i	$-0.208209\pi$
$37$	489027.			1.58718			0.793591	+	0.608451i	$0.208209\pi$
$38$	84966.0	−	147165.i	0.251190	−	0.435074i
$39$		0			0
$40$	−49559.9	−	85840.2i	−0.122439	−	0.212071i
$41$	−295318.	−	511505.i	−0.669184	−	1.15906i	−0.978133	−	0.207982i	$-0.933310\pi$
$41$	−295318.	−	511505.i	−0.669184	−	1.15906i	0.308948	−	0.951079i	$-0.400023\pi$
$42$		0			0
$43$	421321.	−	729750.i	0.808117	−	1.39970i	−0.106050	−	0.994361i	$-0.533820\pi$
$43$	421321.	−	729750.i	0.808117	−	1.39970i	0.914167	−	0.405338i	$-0.132846\pi$
$44$	−348978.			−0.617609
$45$		0			0
$46$	390859.			0.592062
$47$	−613184.	+	1.06207e6i	−0.861486	+	1.49214i	0.00900942	+	0.999959i	$0.497132\pi$
$47$	−613184.	+	1.06207e6i	−0.861486	+	1.49214i	−0.870495	+	0.492177i	$0.836201\pi$
$48$		0			0
$49$	139151.	+	241016.i	0.168966	+	0.292658i
$50$	−280172.	−	485272.i	−0.316978	−	0.549022i
$51$		0			0
$52$	209869.	−	363504.i	0.206984	−	0.358507i
$53$	958904.			0.884727			0.442364	−	0.896836i	$-0.354140\pi$
$53$	958904.			0.884727			0.442364	+	0.896836i	$0.354140\pi$
$54$		0			0
$55$	326517.			0.264628
$56$	556047.	−	963101.i	0.423110	−	0.732848i
$57$		0			0
$58$	−260057.	−	450432.i	−0.175013	−	0.303132i
$59$	158135.	+	273897.i	0.100241	+	0.173622i	0.911784	−	0.410670i	$-0.134705\pi$
$59$	158135.	+	273897.i	0.100241	+	0.173622i	−0.811543	+	0.584293i	$0.801372\pi$
$60$		0			0
$61$	14861.5	−	25740.8i	0.00838315	−	0.0145200i	−0.861803	−	0.507242i	$-0.830665\pi$
$61$	14861.5	−	25740.8i	0.00838315	−	0.0145200i	0.870187	+	0.492722i	$0.163998\pi$
$62$	−1.14398e6			−0.609604
$63$		0			0
$64$	1.63495e6			0.779604
$65$	−196362.	+	340108.i	−0.0886870	+	0.153610i
$66$		0			0
$67$	−146512.	−	253767.i	−0.0595130	−	0.103080i	0.834734	−	0.550654i	$-0.185621\pi$
$67$	−146512.	−	253767.i	−0.0595130	−	0.103080i	−0.894247	+	0.447574i	$0.852288\pi$
$68$	1.28905e6	+	2.23270e6i	0.497150	+	0.861089i
$69$		0			0
$70$	−184507.	+	319576.i	−0.0642939	+	0.111360i
$71$	714537.			0.236930			0.118465	−	0.992958i	$-0.462203\pi$
$71$	714537.			0.236930			0.118465	+	0.992958i	$0.462203\pi$
$72$		0			0
$73$	−3.96273e6			−1.19224			−0.596121	−	0.802894i	$-0.703292\pi$
$73$	−3.96273e6			−1.19224			−0.596121	+	0.802894i	$0.703292\pi$
$74$	1.85669e6	−	3.21587e6i	0.532632	−	0.922546i
$75$		0			0
$76$	787073.	+	1.36325e6i	0.205668	+	0.356228i
$77$	1.83171e6	+	3.17262e6i	0.457235	+	0.791954i
$78$		0			0
$79$	−1.26902e6	+	2.19801e6i	−0.289584	+	0.501574i	−0.973710	−	0.227789i	$-0.926850\pi$
$79$	−1.26902e6	+	2.19801e6i	−0.289584	+	0.501574i	0.684126	+	0.729363i	$0.260184\pi$
$80$	−160100.			−0.0349603
$81$		0			0
$82$	−4.48492e6			−0.898269
$83$	−831556.	+	1.44030e6i	−0.159631	+	0.276490i	−0.934736	−	0.355344i	$-0.884364\pi$
$83$	−831556.	+	1.44030e6i	−0.159631	+	0.276490i	0.775104	+	0.631833i	$0.217697\pi$
$84$		0			0
$85$	−1.20608e6	−	2.08899e6i	−0.213015	−	0.368953i
$86$	−3.19926e6	−	5.54128e6i	−0.542381	−	0.939432i
$87$		0			0
$88$	−3.73602e6	+	6.47098e6i	−0.584413	+	1.01223i
$89$	4.64819e6			0.698906			0.349453	−	0.936954i	$-0.386368\pi$
$89$	4.64819e6			0.698906			0.349453	+	0.936954i	$0.386368\pi$
$90$		0			0
$91$	−4.40624e6			−0.612947
$92$	−1.81034e6	+	3.13560e6i	−0.242383	+	0.419820i
$93$		0			0
$94$	4.65614e6	+	8.06467e6i	0.578201	+	1.00147i
$95$	−736415.	−	1.27551e6i	−0.0881232	−	0.152634i
$96$		0			0
$97$	−7.35048e6	+	1.27314e7i	−0.817738	+	1.41636i	0.0896064	+	0.995977i	$0.471439\pi$
$97$	−7.35048e6	+	1.27314e7i	−0.817738	+	1.41636i	−0.907345	+	0.420387i	$0.861894\pi$
$98$	2.11325e6			0.226809
$99$		0			0
$100$	5.19068e6			0.519068
$101$	−4.64190e6	+	8.04000e6i	−0.448302	+	0.776482i	−0.998276	−	0.0587000i	$-0.981304\pi$
$101$	−4.64190e6	+	8.04000e6i	−0.448302	+	0.776482i	0.549973	+	0.835182i	$0.314638\pi$
$102$		0			0
$103$	−2.05525e6	−	3.55979e6i	−0.185325	−	0.320992i	0.758361	−	0.651835i	$-0.226000\pi$
$103$	−2.05525e6	−	3.55979e6i	−0.185325	−	0.320992i	−0.943686	+	0.330843i	$0.892667\pi$
$104$	−4.49356e6	−	7.78307e6i	−0.391718	−	0.678476i
$105$		0			0
$106$	3.64066e6	−	6.30581e6i	0.296900	−	0.514246i
$107$	−1.77931e7			−1.40414			−0.702068	−	0.712110i	$-0.747740\pi$
$107$	−1.77931e7			−1.40414			−0.702068	+	0.712110i	$0.747740\pi$
$108$		0			0
$109$	−1.72244e7			−1.27394			−0.636972	−	0.770887i	$-0.719813\pi$
$109$	−1.72244e7			−1.27394			−0.636972	+	0.770887i	$0.719813\pi$
$110$	1.23968e6	−	2.14720e6i	0.0888049	−	0.153815i
$111$		0			0
$112$	−898135.	−	1.55562e6i	−0.0604058	−	0.104626i
$113$	1.08376e6	+	1.87712e6i	0.0706573	+	0.122382i	0.899190	−	0.437559i	$-0.144157\pi$
$113$	1.08376e6	+	1.87712e6i	0.0706573	+	0.122382i	−0.828532	+	0.559941i	$0.810824\pi$
$114$		0			0
$115$	1.69382e6	−	2.93378e6i	0.103854	−	0.179881i
$116$	4.81802e6			0.286593
$117$		0			0
$118$	2.40156e6			0.134557
$119$	1.35319e7	−	2.34379e7i	0.736112	−	1.27498i
$120$		0			0
$121$	−2.56349e6	−	4.44009e6i	−0.131547	−	0.227847i
$122$	−112849.	−	195460.i	−0.00562649	−	0.00974537i
$123$		0			0
$124$	5.29857e6	−	9.17739e6i	0.249564	−	0.432258i
$125$	−9.99825e6			−0.457867
$126$		0			0
$127$	9.66827e6			0.418828			0.209414	−	0.977827i	$-0.432844\pi$
$127$	9.66827e6			0.418828			0.209414	+	0.977827i	$0.432844\pi$
$128$	−4.94817e6	+	8.57048e6i	−0.208550	+	0.361219i
$129$		0			0
$130$	1.49105e6	+	2.58257e6i	0.0595238	+	0.103098i
$131$	2.11215e7	+	3.65836e7i	0.820873	+	1.42179i	0.905033	+	0.425342i	$0.139846\pi$
$131$	2.11215e7	+	3.65836e7i	0.820873	+	1.42179i	−0.0841598	+	0.996452i	$0.526821\pi$
$132$		0			0
$133$	8.26236e6	−	1.43108e7i	0.304525	−	0.527453i
$134$	−2.22505e6			−0.0798864
$135$		0			0
$136$	5.52002e7			1.88172
$137$	4.00725e6	−	6.94077e6i	0.133145	−	0.230614i	−0.791742	−	0.610855i	$-0.790826\pi$
$137$	4.00725e6	−	6.94077e6i	0.133145	−	0.230614i	0.924887	+	0.380241i	$0.124159\pi$
$138$		0			0
$139$	−7.30050e6	−	1.26448e7i	−0.230569	−	0.399357i	0.727407	−	0.686207i	$-0.240725\pi$
$139$	−7.30050e6	−	1.26448e7i	−0.230569	−	0.399357i	−0.957976	+	0.286849i	$0.907392\pi$
$140$	−1.70916e6	−	2.96035e6i	−0.0526423	−	0.0911791i
$141$		0			0
$142$	2.71288e6	−	4.69884e6i	0.0795099	−	0.137715i
$143$	2.96050e7			0.846622
$144$		0			0
$145$	−4.50792e6			−0.122797
$146$	−1.50453e7	+	2.60592e7i	−0.400097	+	0.692988i
$147$		0			0
$148$	1.71992e7	+	2.97899e7i	0.436106	+	0.755358i
$149$	−2.03543e7	−	3.52547e7i	−0.504086	−	0.873103i	−0.999989	−	0.00472499i	$-0.998496\pi$
$149$	−2.03543e7	−	3.52547e7i	−0.504086	−	0.873103i	0.495902	−	0.868378i	$-0.334837\pi$
$150$		0			0
$151$	2.64970e7	−	4.58941e7i	0.626293	−	1.08477i	−0.361997	−	0.932179i	$-0.617905\pi$
$151$	2.64970e7	−	4.58941e7i	0.626293	−	1.08477i	0.988289	−	0.152591i	$-0.0487618\pi$
$152$	3.37044e7			0.778456
$153$		0			0
$154$	2.78178e7			0.613762
$155$	−4.95754e6	+	8.58671e6i	−0.106931	+	0.185211i
$156$		0			0
$157$	−2.05335e7	−	3.55650e7i	−0.423461	−	0.733456i	0.572814	−	0.819685i	$-0.305852\pi$
$157$	−2.05335e7	−	3.55650e7i	−0.423461	−	0.733456i	−0.996275	+	0.0862290i	$0.972518\pi$
$158$	9.63618e6	+	1.66904e7i	0.194359	+	0.336640i
$159$		0			0
$160$	5.73581e6	−	9.93472e6i	0.110707	−	0.191750i
$161$	3.80083e7			0.717775
$162$		0			0
$163$	−4.33772e7			−0.784522			−0.392261	−	0.919854i	$-0.628307\pi$
$163$	−4.33772e7			−0.784522			−0.392261	+	0.919854i	$0.628307\pi$
$164$	2.07728e7	−	3.59795e7i	0.367740	−	0.636945i
$165$		0			0
$166$	6.31433e6	+	1.09367e7i	0.107139	+	0.185571i
$167$	−1.93422e7	−	3.35017e7i	−0.321365	−	0.556621i	0.659405	−	0.751788i	$-0.270808\pi$
$167$	−1.93422e7	−	3.35017e7i	−0.321365	−	0.556621i	−0.980770	+	0.195167i	$0.937475\pi$
$168$		0			0
$169$	1.35703e7	−	2.35044e7i	0.216265	−	0.374582i
$170$	−1.83165e7			−0.285937
$171$		0			0
$172$	5.92719e7			0.888177
$173$	−2.65274e7	+	4.59467e7i	−0.389523	+	0.674673i	−0.992385	−	0.123172i	$-0.960693\pi$
$173$	−2.65274e7	+	4.59467e7i	−0.389523	+	0.674673i	0.602863	+	0.797845i	$0.294027\pi$
$174$		0			0
$175$	−2.72448e7	−	4.71893e7i	−0.384282	−	0.665596i
$176$	6.03447e6	+	1.04520e7i	0.0834344	+	0.144513i
$177$		0			0
$178$	1.76477e7	−	3.05668e7i	0.234541	−	0.406237i
$179$	−1.34132e8			−1.74802			−0.874012	−	0.485904i	$-0.838491\pi$
$179$	−1.34132e8			−1.74802			−0.874012	+	0.485904i	$0.838491\pi$
$180$		0			0
$181$	6.35105e6			0.0796105			0.0398053	−	0.999207i	$-0.487326\pi$
$181$	6.35105e6			0.0796105			0.0398053	+	0.999207i	$0.487326\pi$
$182$	−1.67291e7	+	2.89757e7i	−0.205695	+	0.356274i
$183$		0			0
$184$	3.87616e7	+	6.71370e7i	0.458711	+	0.794510i
$185$	−1.60922e7	−	2.78726e7i	−0.186859	−	0.323650i
$186$		0			0
$187$	−9.09193e7	+	1.57477e8i	−1.01674	+	1.76105i
$188$	−8.62633e7			−0.946833
$189$		0			0
$190$	−1.11838e7			−0.118291
$191$	5.79216e7	−	1.00323e8i	0.601483	−	1.04180i	−0.391113	−	0.920343i	$-0.627910\pi$
$191$	5.79216e7	−	1.00323e8i	0.601483	−	1.04180i	0.992597	−	0.121457i	$-0.0387568\pi$
$192$		0			0
$193$	−2.86928e7	−	4.96974e7i	−0.287291	−	0.497603i	0.685871	−	0.727723i	$-0.259421\pi$
$193$	−2.86928e7	−	4.96974e7i	−0.287291	−	0.497603i	−0.973162	+	0.230120i	$0.926088\pi$
$194$	5.58150e7	+	9.66744e7i	0.548839	+	0.950617i
$195$		0			0
$196$	−9.78793e6	+	1.69532e7i	−0.0928527	+	0.160826i
$197$	1.04846e8			0.977059			0.488530	−	0.872547i	$-0.337533\pi$
$197$	1.04846e8			0.977059			0.488530	+	0.872547i	$0.337533\pi$
$198$		0			0
$199$	−2.10623e8			−1.89461			−0.947304	−	0.320335i	$-0.896205\pi$
$199$	−2.10623e8			−1.89461			−0.947304	+	0.320335i	$0.896205\pi$
$200$	5.55694e7	−	9.62490e7i	0.491169	−	0.850729i
$201$		0			0
$202$	3.52477e7	+	6.10509e7i	0.300886	+	0.521149i
$203$	−2.52888e7	−	4.38015e7i	−0.212174	−	0.367496i
$204$		0			0
$205$	−1.94358e7	+	3.36638e7i	−0.157567	+	0.272913i
$206$	−3.12125e7			−0.248768
$207$		0			0
$208$	−1.45161e7			−0.111848
$209$	−5.55139e7	+	9.61529e7i	−0.420620	+	0.728535i
$210$		0			0
$211$	3.24596e7	+	5.62218e7i	0.237878	+	0.412018i	0.960105	−	0.279639i	$-0.0902147\pi$
$211$	3.24596e7	+	5.62218e7i	0.237878	+	0.412018i	−0.722227	+	0.691656i	$0.756881\pi$
$212$	3.37249e7	+	5.84132e7i	0.243094	+	0.421052i
$213$		0			0
$214$	−6.75551e7	+	1.17009e8i	−0.471205	+	0.816151i
$215$	−5.54570e7			−0.380559
$216$		0			0
$217$	−1.11244e8			−0.739041
$218$	−6.53956e7	+	1.13269e8i	−0.427515	+	0.740477i
$219$		0			0
$220$	1.14837e7	+	1.98903e7i	0.0727112	+	0.125940i
$221$	−1.09355e8	−	1.89408e8i	−0.681497	−	1.18039i
$222$		0			0
$223$	−7.04395e7	+	1.22005e8i	−0.425352	+	0.736732i	−0.996453	−	0.0841479i	$-0.973183\pi$
$223$	−7.04395e7	+	1.22005e8i	−0.425352	+	0.736732i	0.571101	+	0.820880i	$0.306517\pi$
$224$	1.28708e8			0.765135
$225$		0			0
$226$	1.64588e7			0.0948457
$227$	1.70081e6	−	2.94589e6i	0.00965086	−	0.0167158i	−0.861160	−	0.508335i	$-0.830261\pi$
$227$	1.70081e6	−	2.94589e6i	0.00965086	−	0.0167158i	0.870811	+	0.491619i	$0.163595\pi$
$228$		0			0
$229$	1.38084e7	+	2.39169e7i	0.0759836	+	0.131608i	0.901514	−	0.432751i	$-0.142457\pi$
$229$	1.38084e7	+	2.39169e7i	0.0759836	+	0.131608i	−0.825530	+	0.564358i	$0.809124\pi$
$230$	−1.28618e7	−	2.22774e7i	−0.0697037	−	0.120730i
$231$		0			0
$232$	5.15799e7	−	8.93390e7i	0.271189	−	0.469713i
$233$	3.51458e8			1.82023			0.910117	−	0.414351i	$-0.135992\pi$
$233$	3.51458e8			1.82023			0.910117	+	0.414351i	$0.135992\pi$
$234$		0			0
$235$	8.07112e7			0.405692
$236$	−1.11233e7	+	1.92661e7i	−0.0550859	+	0.0954116i
$237$		0			0
$238$	−1.02753e8	−	1.77973e8i	−0.494054	−	0.855726i
$239$	1.89598e8	+	3.28393e8i	0.898339	+	1.55597i	0.829617	+	0.558334i	$0.188559\pi$
$239$	1.89598e8	+	3.28393e8i	0.898339	+	1.55597i	0.0687228	+	0.997636i	$0.478108\pi$
$240$		0			0
$241$	−1.49638e7	+	2.59180e7i	−0.0688624	+	0.119273i	−0.898401	−	0.439177i	$-0.855270\pi$
$241$	−1.49638e7	+	2.59180e7i	−0.0688624	+	0.119273i	0.829538	+	0.558450i	$0.188604\pi$
$242$	−3.89311e7			−0.176581
$243$		0			0
$244$	2.09072e6			0.00921367
$245$	9.15796e6	−	1.58620e7i	0.0397848	−	0.0689093i
$246$		0			0
$247$	−6.67702e7	−	1.15649e8i	−0.281932	−	0.488320i
$248$	−1.13449e8	−	1.96499e8i	−0.472301	−	0.818050i
$249$		0			0
$250$	−3.79603e7	+	6.57492e7i	−0.153653	+	0.266134i
$251$	−1.71456e7			−0.0684374			−0.0342187	−	0.999414i	$-0.510894\pi$
$251$	−1.71456e7			−0.0684374			−0.0342187	+	0.999414i	$0.510894\pi$
$252$		0			0
$253$	−2.55374e8			−0.991414
$254$	3.67074e7	−	6.35792e7i	0.140552	−	0.243443i
$255$		0			0
$256$	1.42210e8	+	2.46315e8i	0.529774	+	0.917595i
$257$	4.89725e7	+	8.48228e7i	0.179964	+	0.311707i	0.941868	−	0.335983i	$-0.109068\pi$
$257$	4.89725e7	+	8.48228e7i	0.179964	+	0.311707i	−0.761904	+	0.647690i	$0.775735\pi$
$258$		0			0
$259$	1.80550e8	−	3.12722e8i	0.645726	−	1.11843i
$260$	−2.76243e7			−0.0974732
$261$		0			0
$262$	3.20768e8			1.10189
$263$	4.46270e7	−	7.72962e7i	0.151270	−	0.262007i	−0.780425	−	0.625250i	$-0.784997\pi$
$263$	4.46270e7	−	7.72962e7i	0.151270	−	0.262007i	0.931695	+	0.363243i	$0.118330\pi$
$264$		0			0
$265$	−3.15543e7	−	5.46536e7i	−0.104159	−	0.180409i
$266$	−6.27393e7	−	1.08668e8i	−0.204387	−	0.354009i
$267$		0			0
$268$	1.03057e7	−	1.78501e7i	0.0327045	−	0.0566459i
$269$	−5.34653e8			−1.67471			−0.837354	−	0.546662i	$-0.815898\pi$
$269$	−5.34653e8			−1.67471			−0.837354	+	0.546662i	$0.815898\pi$
$270$		0			0
$271$	2.11501e8			0.645535			0.322768	−	0.946478i	$-0.395387\pi$
$271$	2.11501e8			0.645535			0.322768	+	0.946478i	$0.395387\pi$
$272$	4.45800e7	−	7.72149e7i	0.134323	−	0.232654i
$273$		0			0
$274$	−3.04286e7	−	5.27039e7i	−0.0893625	−	0.154780i
$275$	1.83055e8	+	3.17060e8i	0.530783	+	0.919343i
$276$		0			0
$277$	1.62009e8	−	2.80607e8i	0.457993	−	0.793267i	−0.540862	−	0.841111i	$-0.681902\pi$
$277$	1.62009e8	−	2.80607e8i	0.457993	−	0.793267i	0.998855	+	0.0478444i	$0.0152352\pi$
$278$	−1.10871e8			−0.309501
$279$		0			0
$280$	−7.31905e7			−0.199251
$281$	−2.45605e7	+	4.25400e7i	−0.0660336	+	0.114373i	−0.897152	−	0.441722i	$-0.854368\pi$
$281$	−2.45605e7	+	4.25400e7i	−0.0660336	+	0.114373i	0.831118	+	0.556095i	$0.187701\pi$
$282$		0			0
$283$	2.01509e8	+	3.49023e8i	0.528495	+	0.915381i	0.999448	+	0.0332225i	$0.0105770\pi$
$283$	2.01509e8	+	3.49023e8i	0.528495	+	0.915381i	−0.470952	+	0.882159i	$0.656090\pi$
$284$	2.51304e7	+	4.35272e7i	0.0651008	+	0.112758i
$285$		0			0
$286$	1.12401e8	−	1.94685e8i	0.284112	−	0.492097i
$287$	−4.36128e8			−1.08900
$288$		0			0
$289$	9.33004e8			2.27374
$290$	−1.71152e7	+	2.96444e7i	−0.0412087	+	0.0713756i
$291$		0			0
$292$	−1.39370e8	−	2.41396e8i	−0.327590	−	0.567402i
$293$	−2.42953e8	−	4.20807e8i	−0.564268	−	0.977340i	−0.997117	−	0.0758743i	$-0.975825\pi$
$293$	−2.42953e8	−	4.20807e8i	−0.564268	−	0.977340i	0.432850	−	0.901466i	$-0.357508\pi$
$294$		0			0
$295$	1.04074e7	−	1.80261e7i	0.0236028	−	0.0408812i
$296$	7.36512e8			1.65066
$297$		0			0
$298$	−3.09117e8			−0.676652
$299$	1.53578e8	−	2.66004e8i	0.332260	−	0.575492i
$300$		0			0
$301$	−3.11106e8	−	5.38851e8i	−0.657545	−	1.13890i
$302$	−2.01202e8	−	3.48492e8i	−0.420347	−	0.728062i
$303$		0			0
$304$	2.72199e7	−	4.71463e7i	0.0555686	−	0.0962476i
$305$	−1.95616e6			−0.00394780
$306$		0			0
$307$	2.82215e8			0.556668			0.278334	−	0.960484i	$-0.410218\pi$
$307$	2.82215e8			0.556668			0.278334	+	0.960484i	$0.410218\pi$
$308$	−1.28843e8	+	2.23163e8i	−0.251267	+	0.435206i
$309$		0			0
$310$	3.76445e7	+	6.52022e7i	0.0717688	+	0.124307i
$311$	−3.52220e7	−	6.10063e7i	−0.0663977	−	0.115004i	0.830915	−	0.556399i	$-0.187817\pi$
$311$	−3.52220e7	−	6.10063e7i	−0.0663977	−	0.115004i	−0.897313	+	0.441395i	$0.854484\pi$
$312$		0			0
$313$	2.39048e8	−	4.14044e8i	0.440636	−	0.763205i	−0.557100	−	0.830445i	$-0.688086\pi$
$313$	2.39048e8	−	4.14044e8i	0.440636	−	0.763205i	0.997737	+	0.0672406i	$0.0214195\pi$
$314$	−3.11837e8			−0.568426
$315$		0			0
$316$	−1.78527e8			−0.318273
$317$	−3.44735e8	+	5.97099e8i	−0.607824	+	1.05278i	0.383774	+	0.923427i	$0.374624\pi$
$317$	−3.44735e8	+	5.97099e8i	−0.607824	+	1.05278i	−0.991598	+	0.129355i	$0.958709\pi$
$318$		0			0
$319$	1.69913e8	+	2.94297e8i	0.293061	+	0.507597i
$320$	−5.38006e7	−	9.31854e7i	−0.0917830	−	0.158973i
$321$		0			0
$322$	1.44306e8	−	2.49945e8i	0.240873	−	0.417205i
$323$	8.20225e8			1.35433
$324$		0			0
$325$	−4.40344e8			−0.711542
$326$	−1.64690e8	+	2.85251e8i	−0.263273	+	0.456002i
$327$		0			0
$328$	−4.44771e8	−	7.70365e8i	−0.695949	−	1.20542i
$329$	4.52778e8	+	7.84234e8i	0.700970	+	1.21412i
$330$		0			0
$331$	−2.68417e8	+	4.64911e8i	−0.406829	+	0.704648i	−0.994532	−	0.104428i	$-0.966699\pi$
$331$	−2.68417e8	+	4.64911e8i	−0.406829	+	0.704648i	0.587704	+	0.809076i	$0.300032\pi$
$332$	−1.16984e8			−0.175446
$333$		0			0
$334$	−2.93746e8			−0.431379
$335$	−9.64244e6	+	1.67012e7i	−0.0140130	+	0.0242712i
$336$		0			0
$337$	6.20056e8	+	1.07397e9i	0.882524	+	1.52858i	0.848526	+	0.529154i	$0.177491\pi$
$337$	6.20056e8	+	1.07397e9i	0.882524	+	1.52858i	0.0339983	+	0.999422i	$0.489176\pi$
$338$	−1.03044e8	−	1.78478e8i	−0.145150	−	0.251407i
$339$		0			0
$340$	8.48364e7	−	1.46941e8i	0.117059	−	0.202753i
$341$	7.47438e8			1.02079
$342$		0			0
$343$	8.13607e8			1.08864
$344$	6.34542e8	−	1.09906e9i	0.840438	−	1.45568i
$345$		0			0
$346$	2.01433e8	+	3.48891e8i	0.261435	+	0.452818i
$347$	−2.43383e7	−	4.21552e7i	−0.0312706	−	0.0541623i	0.849967	−	0.526837i	$-0.176622\pi$
$347$	−2.43383e7	−	4.21552e7i	−0.0312706	−	0.0541623i	−0.881237	+	0.472674i	$0.843289\pi$
$348$		0			0
$349$	−6.18327e8	+	1.07097e9i	−0.778626	+	1.34862i	0.154107	+	0.988054i	$0.450750\pi$
$349$	−6.18327e8	+	1.07097e9i	−0.778626	+	1.34862i	−0.932734	+	0.360566i	$0.882584\pi$
$350$	−4.13760e8			−0.515835
$351$		0			0
$352$	−8.64777e8			−1.05683
$353$	−3.03429e7	+	5.25555e7i	−0.0367152	+	0.0635926i	−0.883799	−	0.467867i	$-0.845023\pi$
$353$	−3.03429e7	+	5.25555e7i	−0.0367152	+	0.0635926i	0.847084	+	0.531459i	$0.178356\pi$
$354$		0			0
$355$	−2.35130e7	−	4.07257e7i	−0.0278939	−	0.0483136i
$356$	1.63478e8	+	2.83152e8i	0.192037	+	0.332617i
$357$		0			0
$358$	−5.09259e8	+	8.82062e8i	−0.586608	+	1.01604i
$359$	−5.36061e8			−0.611482			−0.305741	−	0.952115i	$-0.598904\pi$
$359$	−5.36061e8			−0.611482			−0.305741	+	0.952115i	$0.598904\pi$
$360$		0			0
$361$	−3.93055e8			−0.439722
$362$	2.41130e7	−	4.17649e7i	0.0267160	−	0.0462734i
$363$		0			0
$364$	−1.54968e8	−	2.68413e8i	−0.168418	−	0.291708i
$365$	1.30400e8	+	2.25860e8i	0.140363	+	0.243116i
$366$		0			0
$367$	2.15664e8	−	3.73540e8i	0.227743	−	0.394463i	−0.729396	−	0.684092i	$-0.760199\pi$
$367$	2.15664e8	−	3.73540e8i	0.227743	−	0.394463i	0.957139	+	0.289629i	$0.0935319\pi$
$368$	1.25216e8			0.130977
$369$		0			0
$370$	−2.44389e8			−0.250828
$371$	3.54030e8	−	6.13197e8i	0.359941	−	0.623435i
$372$		0			0
$373$	−1.34844e8	−	2.33557e8i	−0.134540	−	0.233030i	0.790882	−	0.611969i	$-0.209622\pi$
$373$	−1.34844e8	−	2.33557e8i	−0.134540	−	0.233030i	−0.925422	+	0.378939i	$0.876289\pi$
$374$	6.90385e8	+	1.19578e9i	0.682403	+	1.18196i
$375$		0			0
$376$	−9.23501e8	+	1.59955e9i	−0.895942	+	1.55182i
$377$	−4.08730e8			−0.392864
$378$		0			0
$379$	−8.03807e8			−0.758429			−0.379214	−	0.925309i	$-0.623806\pi$
$379$	−8.03807e8			−0.758429			−0.379214	+	0.925309i	$0.623806\pi$
$380$	5.17998e7	−	8.97199e7i	0.0484268	−	0.0838776i
$381$		0			0
$382$	−4.39821e8	−	7.61792e8i	−0.403696	−	0.699222i
$383$	−4.91962e8	−	8.52103e8i	−0.447441	−	0.774990i	0.550778	−	0.834652i	$-0.314331\pi$
$383$	−4.91962e8	−	8.52103e8i	−0.447441	−	0.774990i	−0.998219	+	0.0596617i	$0.980998\pi$
$384$		0			0
$385$	1.20551e8	−	2.08800e8i	0.107661	−	0.186474i
$386$	−4.35751e8			−0.385641
$387$		0			0
$388$	−1.03407e9			−0.898752
$389$	1.08125e8	−	1.87278e8i	0.0931326	−	0.161310i	−0.815695	−	0.578482i	$-0.803645\pi$
$389$	1.08125e8	−	1.87278e8i	0.0931326	−	0.161310i	0.908828	+	0.417172i	$0.136979\pi$
$390$		0			0
$391$	9.43296e8	+	1.63384e9i	0.798048	+	1.38226i
$392$	2.09571e8	+	3.62988e8i	0.175724	+	0.304363i
$393$		0			0
$394$	3.98069e8	−	6.89475e8i	0.327885	−	0.567913i
$395$	1.67037e8			0.136371
$396$		0			0
$397$	4.33533e8			0.347741			0.173870	−	0.984769i	$-0.444373\pi$
$397$	4.33533e8			0.347741			0.173870	+	0.984769i	$0.444373\pi$
$398$	−7.99670e8	+	1.38507e9i	−0.635799	+	1.10124i
$399$		0			0
$400$	−8.97565e7	−	1.55463e8i	−0.0701223	−	0.121455i
$401$	−1.21357e9	−	2.10196e9i	−0.939849	−	1.62787i	−0.765751	−	0.643137i	$-0.777632\pi$
$401$	−1.21357e9	−	2.10196e9i	−0.939849	−	1.62787i	−0.174098	−	0.984728i	$-0.555701\pi$
$402$		0			0
$403$	−4.49497e8	+	7.78551e8i	−0.342104	+	0.592542i
$404$	−6.53027e8			−0.492715
$405$		0			0
$406$	−3.84055e8			−0.284808
$407$	−1.21310e9	+	2.10114e9i	−0.891897	+	1.54481i
$408$		0			0
$409$	−4.59488e8	−	7.95857e8i	−0.332080	−	0.575179i	0.650839	−	0.759215i	$-0.274417\pi$
$409$	−4.59488e8	−	7.95857e8i	−0.332080	−	0.575179i	−0.982920	+	0.184036i	$0.941084\pi$
$410$	1.47584e8	+	2.55622e8i	0.105753	+	0.183170i
$411$		0			0
$412$	1.44567e8	−	2.50397e8i	0.101842	−	0.176396i
$413$	2.33535e8			0.163127
$414$		0			0
$415$	1.09455e8			0.0751738
$416$	5.20062e8	−	9.00774e8i	0.354184	−	0.613464i
$417$		0			0
$418$	4.21539e8	+	7.30126e8i	0.282306	+	0.488969i
$419$	−5.50120e8	−	9.52836e8i	−0.365349	−	0.632804i	0.623483	−	0.781837i	$-0.285717\pi$
$419$	−5.50120e8	−	9.52836e8i	−0.365349	−	0.632804i	−0.988832	+	0.149033i	$0.952384\pi$
$420$		0			0
$421$	1.07454e9	−	1.86116e9i	0.701837	−	1.21562i	−0.265984	−	0.963978i	$-0.585697\pi$
$421$	1.07454e9	−	1.86116e9i	0.701837	−	1.21562i	0.967821	−	0.251640i	$-0.0809699\pi$
$422$	4.92957e8			0.319312
$423$		0			0
$424$	1.44418e9			0.920113
$425$	1.35233e9	−	2.34230e9i	0.854518	−	1.48007i
$426$		0			0
$427$	−1.09738e7	−	1.90071e7i	−0.00682116	−	0.0118146i
$428$	−6.25789e8	−	1.08390e9i	−0.385811	−	0.668244i
$429$		0			0
$430$	−2.10553e8	+	3.64689e8i	−0.127709	+	0.221199i
$431$	1.61850e9			0.973737			0.486869	−	0.873475i	$-0.338139\pi$
$431$	1.61850e9			0.973737			0.486869	+	0.873475i	$0.338139\pi$
$432$		0			0
$433$	1.16527e9			0.689794			0.344897	−	0.938641i	$-0.387914\pi$
$433$	1.16527e9			0.689794			0.344897	+	0.938641i	$0.387914\pi$
$434$	−4.22360e8	+	7.31549e8i	−0.248010	+	0.429566i
$435$		0			0
$436$	−6.05785e8	−	1.04925e9i	−0.350039	−	0.606285i
$437$	5.75962e8	+	9.97596e8i	0.330148	+	0.571834i
$438$		0			0
$439$	−1.04048e9	+	1.80216e9i	−0.586958	+	1.01664i	0.407670	+	0.913129i	$0.366341\pi$
$439$	−1.04048e9	+	1.80216e9i	−0.586958	+	1.01664i	−0.994628	+	0.103512i	$0.966992\pi$
$440$	4.91759e8			0.275212
$441$		0			0
$442$	−1.66074e9			−0.914796
$443$	−1.11550e9	+	1.93210e9i	−0.609616	+	1.05589i	0.381687	+	0.924292i	$0.375343\pi$
$443$	−1.11550e9	+	1.93210e9i	−0.609616	+	1.05589i	−0.991304	+	0.131595i	$0.957990\pi$
$444$		0			0
$445$	−1.52956e8	−	2.64928e8i	−0.0822824	−	0.142517i
$446$	5.34874e8	+	9.26429e8i	0.285482	+	0.494470i
$447$		0			0
$448$	6.03627e8	−	1.04551e9i	0.317173	−	0.549359i
$449$	−2.52631e9			−1.31712			−0.658559	−	0.752529i	$-0.728834\pi$
$449$	−2.52631e9			−1.31712			−0.658559	+	0.752529i	$0.728834\pi$
$450$		0			0
$451$	2.93030e9			1.50416
$452$	−7.62319e7	+	1.32038e8i	−0.0388287	+	0.0672532i
$453$		0			0
$454$	−1.29149e7	−	2.23693e7i	−0.00647734	−	0.0112191i
$455$	1.44994e8	+	2.51138e8i	0.0721624	+	0.124989i
$456$		0			0
$457$	9.38136e8	−	1.62490e9i	0.459790	−	0.796379i	−0.539160	−	0.842203i	$-0.681258\pi$
$457$	9.38136e8	−	1.62490e9i	0.459790	−	0.796379i	0.998950	+	0.0458245i	$0.0145915\pi$
$458$	2.09705e8			0.101995
$459$		0			0
$460$	2.38288e8			0.114143
$461$	9.86604e8	−	1.70885e9i	0.469018	−	0.812363i	−0.530355	−	0.847776i	$-0.677941\pi$
$461$	9.86604e8	−	1.70885e9i	0.469018	−	0.812363i	0.999373	+	0.0354126i	$0.0112745\pi$
$462$		0			0
$463$	8.01719e8	+	1.38862e9i	0.375395	+	0.650203i	0.990386	−	0.138331i	$-0.0441737\pi$
$463$	8.01719e8	+	1.38862e9i	0.375395	+	0.650203i	−0.614991	+	0.788534i	$0.710840\pi$
$464$	−8.33126e7	−	1.44302e8i	−0.0387166	−	0.0670592i
$465$		0			0
$466$	1.33438e9	−	2.31121e9i	0.610840	−	1.05801i
$467$	−1.32390e9			−0.601515			−0.300757	−	0.953701i	$-0.597239\pi$
$467$	−1.32390e9			−0.601515			−0.300757	+	0.953701i	$0.597239\pi$
$468$		0			0
$469$	−2.16371e8			−0.0968486
$470$	3.06436e8	−	5.30762e8i	0.136143	−	0.235807i
$471$		0			0
$472$	2.38163e8	+	4.12510e8i	0.104250	+	0.180567i
$473$	2.09029e9	+	3.62048e9i	0.908222	+	1.57309i
$474$		0			0
$475$	8.25712e8	−	1.43017e9i	0.353509	−	0.612296i
$476$	1.90368e9			0.809038
$477$		0			0
$478$	2.87938e9			1.20587
$479$	1.22113e9	−	2.11505e9i	0.507675	−	0.879319i	−0.492285	−	0.870434i	$-0.663838\pi$
$479$	1.22113e9	−	2.11505e9i	0.507675	−	0.879319i	0.999961	−	0.00888546i	$-0.00282837\pi$
$480$		0			0
$481$	−1.45907e9	−	2.52718e9i	−0.597817	−	1.03545i
$482$	1.13626e8	+	1.96806e8i	0.0462181	+	0.0800522i
$483$		0			0
$484$	1.80317e8	−	3.12318e8i	0.0722899	−	0.125210i
$485$	9.67517e8			0.385090
$486$		0			0
$487$	−1.37309e7			−0.00538701			−0.00269350	−	0.999996i	$-0.500857\pi$
$487$	−1.37309e7			−0.00538701			−0.00269350	+	0.999996i	$0.500857\pi$
$488$	2.23825e7	−	3.87676e7i	0.00871845	−	0.0151008i
$489$		0			0
$490$	−6.95399e7	−	1.20447e8i	−0.0267022	−	0.0462496i
$491$	3.16121e8	+	5.47538e8i	0.120522	+	0.208751i	0.919974	−	0.391980i	$-0.128210\pi$
$491$	3.16121e8	+	5.47538e8i	0.120522	+	0.208751i	−0.799451	+	0.600731i	$0.794876\pi$
$492$		0			0
$493$	1.25524e9	−	2.17414e9i	0.471805	−	0.817191i
$494$	−1.01402e9			−0.378446
$495$		0			0
$496$	−3.66488e8			−0.134857
$497$	2.63809e8	−	4.56930e8i	0.0963922	−	0.166956i
$498$		0			0
$499$	1.21683e9	+	2.10761e9i	0.438406	+	0.759342i	0.997567	−	0.0697175i	$-0.0222098\pi$
$499$	1.21683e9	+	2.10761e9i	0.438406	+	0.759342i	−0.559161	+	0.829059i	$0.688876\pi$
$500$	−3.51641e8	−	6.09060e8i	−0.125807	−	0.217904i
$501$		0			0
$502$	−6.50964e7	+	1.12750e8i	−0.0229665	+	0.0397791i
$503$	2.00281e9			0.701700			0.350850	−	0.936432i	$-0.385893\pi$
$503$	2.00281e9			0.701700			0.350850	+	0.936432i	$0.385893\pi$
$504$		0			0
$505$	6.10997e8			0.211115
$506$	−9.69577e8	+	1.67936e9i	−0.332702	+	0.576257i
$507$		0			0
$508$	3.40035e8	+	5.88958e8i	0.115080	+	0.199325i
$509$	2.64155e8	+	4.57529e8i	0.0887863	+	0.153782i	0.906998	−	0.421134i	$-0.138368\pi$
$509$	2.64155e8	+	4.57529e8i	0.0887863	+	0.153782i	−0.818212	+	0.574917i	$0.805035\pi$
$510$		0			0
$511$	−1.46305e9	+	2.53408e9i	−0.485049	+	0.840130i
$512$	8.92980e8			0.294034
$513$		0			0
$514$	7.43734e8			0.241572
$515$	−1.35262e8	+	2.34281e8i	−0.0436367	+	0.0755809i
$516$		0			0
$517$	−3.04217e9	−	5.26919e9i	−0.968202	−	1.67698i
$518$	−1.37099e9	−	2.37462e9i	−0.433390	−	0.750653i
$519$		0			0
$520$	−2.95735e8	+	5.12229e8i	−0.0922341	+	0.159754i
$521$	−1.76283e9			−0.546109			−0.273055	−	0.961999i	$-0.588034\pi$
$521$	−1.76283e9			−0.546109			−0.273055	+	0.961999i	$0.588034\pi$
$522$		0			0
$523$	−3.39318e9			−1.03717			−0.518586	−	0.855026i	$-0.673541\pi$
$523$	−3.39318e9			−1.03717			−0.518586	+	0.855026i	$0.673541\pi$
$524$	−1.48570e9	+	2.57331e9i	−0.451098	+	0.781326i
$525$		0			0
$526$	−3.38870e8	−	5.86940e8i	−0.101527	−	0.175851i
$527$	−2.76087e9	−	4.78197e9i	−0.821693	−	1.42321i
$528$		0			0
$529$	3.77648e8	−	6.54105e8i	0.110915	−	0.192111i
$530$	−4.79208e8			−0.139816
$531$		0			0
$532$	1.16236e9			0.334695
$533$	−1.76223e9	+	3.05227e9i	−0.504101	+	0.873128i
$534$		0			0
$535$	5.85512e8	+	1.01414e9i	0.165309	+	0.286324i
$536$	−2.20659e8	−	3.82192e8i	−0.0618934	−	0.107202i
$537$		0			0
$538$	−2.02991e9	+	3.51591e9i	−0.562004	+	0.973420i
$539$	−1.38073e9			−0.379793
$540$		0			0
$541$	−3.83907e9			−1.04240			−0.521202	−	0.853433i	$-0.674516\pi$
$541$	−3.83907e9			−1.04240			−0.521202	+	0.853433i	$0.674516\pi$
$542$	8.03004e8	−	1.39084e9i	0.216631	−	0.375216i
$543$		0			0
$544$	3.19430e9	+	5.53268e9i	0.850705	+	1.47346i
$545$	5.66796e8	+	9.81719e8i	0.149982	+	0.259776i
$546$		0			0
$547$	−2.15617e9	+	3.73460e9i	−0.563285	+	0.975638i	0.433922	+	0.900950i	$0.357129\pi$
$547$	−2.15617e9	+	3.73460e9i	−0.563285	+	0.975638i	−0.997207	+	0.0746879i	$0.976204\pi$
$548$	5.63744e8			0.146336
$549$		0			0
$550$	2.78001e9			0.712488
$551$	7.66431e8	−	1.32750e9i	0.195183	−	0.338067i
$552$		0			0
$553$	9.37052e8	+	1.62302e9i	0.235627	+	0.408119i
$554$	−1.23019e9	−	2.13076e9i	−0.307390	−	0.532415i
$555$		0			0
$556$	5.13521e8	−	8.89444e8i	0.126706	−	0.219461i
$557$	−1.84109e9			−0.451421			−0.225710	−	0.974194i	$-0.572470\pi$
$557$	−1.84109e9			−0.451421			−0.225710	+	0.974194i	$0.572470\pi$
$558$		0			0
$559$	−5.02825e9			−1.21752
$560$	−5.91092e7	+	1.02380e8i	−0.0142232	+	0.0246353i
$561$		0			0
$562$	1.86497e8	+	3.23023e8i	0.0443195	+	0.0767637i
$563$	5.80275e8	+	1.00507e9i	0.137042	+	0.237364i	0.926376	−	0.376600i	$-0.122907\pi$
$563$	5.80275e8	+	1.00507e9i	0.137042	+	0.237364i	−0.789333	+	0.613965i	$0.789574\pi$
$564$		0			0
$565$	7.13255e7	−	1.23539e8i	0.0166370	−	0.0288161i
$566$	3.06027e9			0.709417
$567$		0			0
$568$	1.07615e9			0.246407
$569$	−3.51460e8	+	6.08746e8i	−0.0799802	+	0.138530i	−0.903241	−	0.429133i	$-0.858819\pi$
$569$	−3.51460e8	+	6.08746e8i	−0.0799802	+	0.138530i	0.823261	+	0.567663i	$0.192152\pi$
$570$		0			0
$571$	3.97964e8	+	6.89295e8i	0.0894577	+	0.154945i	0.907282	−	0.420523i	$-0.138153\pi$
$571$	3.97964e8	+	6.89295e8i	0.0894577	+	0.154945i	−0.817824	+	0.575468i	$0.804820\pi$
$572$	1.04122e9	+	1.80344e9i	0.232624	+	0.402917i
$573$		0			0
$574$	−1.65584e9	+	2.86801e9i	−0.365450	+	0.632978i
$575$	3.79842e9			0.833232
$576$		0			0
$577$	−4.44712e9			−0.963749			−0.481874	−	0.876240i	$-0.660044\pi$
$577$	−4.44712e9			−0.963749			−0.481874	+	0.876240i	$0.660044\pi$
$578$	3.54233e9	−	6.13550e9i	0.763030	−	1.32161i
$579$		0			0
$580$	−1.58545e8	−	2.74608e8i	−0.0337407	−	0.0584406i
$581$	6.14025e8	+	1.06352e9i	0.129888	+	0.224973i
$582$		0			0
$583$	−2.37869e9	+	4.12001e9i	−0.497161	+	0.861109i
$584$	−5.96818e9			−1.23993
$585$		0			0
$586$	−3.68967e9			−0.757436
$587$	−2.75423e9	+	4.77046e9i	−0.562039	+	0.973480i	0.435279	+	0.900295i	$0.356650\pi$
$587$	−2.75423e9	+	4.77046e9i	−0.562039	+	0.973480i	−0.997318	+	0.0731848i	$0.976684\pi$
$588$		0			0
$589$	−1.68575e9	−	2.91980e9i	−0.339930	−	0.588776i
$590$	−7.90271e7	−	1.36879e8i	−0.0158414	−	0.0274381i
$591$		0			0
$592$	5.94812e8	−	1.03025e9i	0.117830	−	0.204087i
$593$	−8.05850e9			−1.58695			−0.793475	−	0.608603i	$-0.791730\pi$
$593$	−8.05850e9			−1.58695			−0.793475	+	0.608603i	$0.791730\pi$
$594$		0			0
$595$	−1.78115e9			−0.346650
$596$	1.43173e9	−	2.47984e9i	0.277013	−	0.479801i
$597$		0			0
$598$	−1.16617e9	−	2.01987e9i	−0.223002	−	0.386251i
$599$	1.69783e9	+	2.94073e9i	0.322775	+	0.559063i	0.981060	−	0.193706i	$-0.0620508\pi$
$599$	1.69783e9	+	2.94073e9i	0.322775	+	0.559063i	−0.658284	+	0.752770i	$0.728717\pi$
$600$		0			0
$601$	−6.75884e8	+	1.17067e9i	−0.127002	+	0.219974i	−0.922514	−	0.385964i	$-0.873869\pi$
$601$	−6.75884e8	+	1.17067e9i	−0.127002	+	0.219974i	0.795512	+	0.605938i	$0.207202\pi$
$602$	−4.72469e9			−0.882645
$603$		0			0
$604$	3.72762e9			0.688339
$605$	−1.68711e8	+	2.92217e8i	−0.0309742	+	0.0536489i
$606$		0			0
$607$	−2.45648e9	−	4.25475e9i	−0.445814	−	0.772172i	0.552295	−	0.833649i	$-0.313752\pi$
$607$	−2.45648e9	−	4.25475e9i	−0.445814	−	0.772172i	−0.998109	+	0.0614769i	$0.980419\pi$
$608$	1.95039e9	+	3.37817e9i	0.351932	+	0.609564i
$609$		0			0
$610$	−7.42695e6	+	1.28638e7i	−0.00132482	+	0.00229465i
$611$	7.31802e9			1.29793
$612$		0			0
$613$	−2.06110e9			−0.361399			−0.180700	−	0.983538i	$-0.557836\pi$
$613$	−2.06110e9			−0.361399			−0.180700	+	0.983538i	$0.557836\pi$
$614$	1.07149e9	−	1.85587e9i	0.186809	−	0.323562i
$615$		0			0
$616$	2.75870e9	+	4.77820e9i	0.475523	+	0.823630i
$617$	1.86139e9	+	3.22402e9i	0.319035	+	0.552585i	0.980287	−	0.197579i	$-0.0633078\pi$
$617$	1.86139e9	+	3.22402e9i	0.319035	+	0.552585i	−0.661252	+	0.750164i	$0.729974\pi$
$618$		0			0
$619$	−1.65033e9	+	2.85845e9i	−0.279675	+	0.484410i	−0.971304	−	0.237842i	$-0.923560\pi$
$619$	−1.65033e9	+	2.85845e9i	−0.279675	+	0.484410i	0.691629	+	0.722253i	$0.256893\pi$
$620$	−6.97432e8			−0.117525
$621$		0			0
$622$	−5.34909e8			−0.0891278
$623$	1.71612e9	−	2.97241e9i	0.284341	−	0.492494i
$624$		0			0
$625$	−2.55355e9	−	4.42289e9i	−0.418374	−	0.724646i
$626$	−1.81519e9	−	3.14399e9i	−0.295741	−	0.512238i
$627$		0			0
$628$	1.44433e9	−	2.50166e9i	0.232707	−	0.403060i
$629$	1.79236e10			2.87177
$630$		0			0
$631$	5.12709e9			0.812397			0.406199	−	0.913785i	$-0.366854\pi$
$631$	5.12709e9			0.812397			0.406199	+	0.913785i	$0.366854\pi$
$632$	−1.91124e9	+	3.31037e9i	−0.301166	+	0.521635i
$633$		0			0
$634$	2.61771e9	+	4.53400e9i	0.407952	+	0.706593i
$635$	−3.18150e8	−	5.51052e8i	−0.0493087	−	0.0854052i
$636$		0			0
$637$	8.30345e8	−	1.43820e9i	0.127283	−	0.220461i
$638$	2.58043e9			0.393386
$639$		0			0
$640$	6.51310e8			0.0982105
$641$	2.25921e9	−	3.91306e9i	0.338808	−	0.586832i	−0.645401	−	0.763844i	$-0.723310\pi$
$641$	2.25921e9	−	3.91306e9i	0.338808	−	0.586832i	0.984209	+	0.177012i	$0.0566431\pi$
$642$		0			0
$643$	4.85300e9	+	8.40564e9i	0.719900	+	1.24690i	0.961039	+	0.276412i	$0.0891455\pi$
$643$	4.85300e9	+	8.40564e9i	0.719900	+	1.24690i	−0.241140	+	0.970490i	$0.577521\pi$
$644$	1.33676e9	+	2.31534e9i	0.197221	+	0.341597i
$645$		0			0
$646$	3.11414e9	−	5.39385e9i	0.454490	−	0.787201i
$647$	−6.12536e9			−0.889132			−0.444566	−	0.895746i	$-0.646642\pi$
$647$	−6.12536e9			−0.889132			−0.444566	+	0.895746i	$0.646642\pi$
$648$		0			0
$649$	−1.56910e9			−0.225317
$650$	−1.67185e9	+	2.89573e9i	−0.238782	+	0.413582i
$651$		0			0
$652$	−1.52559e9	−	2.64239e9i	−0.215561	−	0.373363i
$653$	−6.50516e9	−	1.12673e10i	−0.914243	−	1.58352i	−0.808006	−	0.589175i	$-0.799453\pi$
$653$	−6.50516e9	−	1.12673e10i	−0.914243	−	1.58352i	−0.106237	−	0.994341i	$-0.533880\pi$
$654$		0			0
$655$	1.39008e9	−	2.40768e9i	0.193283	−	0.334776i
$656$	−1.43680e9			−0.198716
$657$		0			0
$658$	6.87623e9			0.940936
$659$	−6.15687e9	+	1.06640e10i	−0.838032	+	1.45151i	0.0535054	+	0.998568i	$0.482961\pi$
$659$	−6.15687e9	+	1.06640e10i	−0.838032	+	1.45151i	−0.891538	+	0.452947i	$0.850373\pi$
$660$		0			0
$661$	−3.48888e9	−	6.04291e9i	−0.469873	−	0.813844i	0.529533	−	0.848289i	$-0.322367\pi$
$661$	−3.48888e9	−	6.04291e9i	−0.469873	−	0.813844i	−0.999407	+	0.0344448i	$0.989034\pi$
$662$	2.03819e9	+	3.53025e9i	0.273050	+	0.472936i
$663$		0			0
$664$	−1.25239e9	+	2.16920e9i	−0.166016	+	0.287548i
$665$	−1.08755e9			−0.143407
$666$		0			0
$667$	3.52572e9			0.460053
$668$	1.36054e9	−	2.35653e9i	0.176601	−	0.305883i
$669$		0			0
$670$	7.32188e7	+	1.26819e8i	0.00940504	+	0.0162900i
$671$	7.37316e7	+	1.27707e8i	0.00942161	+	0.0163187i
$672$		0			0
$673$	5.32311e9	−	9.21990e9i	0.673152	−	1.16593i	−0.303854	−	0.952719i	$-0.598273\pi$
$673$	5.32311e9	−	9.21990e9i	0.673152	−	1.16593i	0.977005	−	0.213214i	$-0.0683933\pi$
$674$	9.41665e9			1.18464
$675$		0			0
$676$	1.90908e9			0.237690
$677$	−2.16674e9	+	3.75291e9i	−0.268378	+	0.464844i	−0.968443	−	0.249235i	$-0.919821\pi$
$677$	−2.16674e9	+	3.75291e9i	−0.268378	+	0.464844i	0.700065	+	0.714079i	$0.253154\pi$
$678$		0			0
$679$	5.42763e9	+	9.40093e9i	0.665374	+	1.15246i
$680$	−1.81645e9	−	3.14618e9i	−0.221535	−	0.383710i
$681$		0			0
$682$	2.83779e9	−	4.91520e9i	0.342559	−	0.593330i
$683$	−6.16980e9			−0.740966			−0.370483	−	0.928839i	$-0.620808\pi$
$683$	−6.16980e9			−0.740966			−0.370483	+	0.928839i	$0.620808\pi$
$684$		0			0
$685$	−5.27461e8			−0.0627008
$686$	3.08902e9	−	5.35033e9i	0.365330	−	0.632771i
$687$		0			0
$688$	−1.02492e9	−	1.77522e9i	−0.119986	−	0.207822i
$689$	−2.86100e9	−	4.95540e9i	−0.333235	−	0.577181i
$690$		0			0
$691$	−3.78637e9	+	6.55818e9i	−0.436565	+	0.756154i	−0.997422	−	0.0717594i	$-0.977139\pi$
$691$	−3.78637e9	+	6.55818e9i	−0.436565	+	0.756154i	0.560856	+	0.827913i	$0.310472\pi$
$692$	−3.73190e9			−0.428113
$693$		0			0
$694$	−3.69620e8			−0.0419756
$695$	−4.80470e8	+	8.32198e8i	−0.0542899	+	0.0940329i
$696$		0			0
$697$	−1.08239e10	−	1.87475e10i	−1.21079	−	2.09715i
$698$	4.69519e9	+	8.13231e9i	0.522588	+	0.905149i
$699$		0			0
$700$	1.91641e9	−	3.31932e9i	0.211176	−	0.365768i
$701$	−4.42696e9			−0.485391			−0.242696	−	0.970102i	$-0.578032\pi$
$701$	−4.42696e9			−0.485391			−0.242696	+	0.970102i	$0.578032\pi$
$702$		0			0
$703$	1.09439e10			1.18803
$704$	−4.05571e9	+	7.02469e9i	−0.438089	+	0.758792i
$705$		0			0
$706$	2.30406e8	+	3.99074e8i	0.0246420	+	0.0426812i
$707$	3.42760e9	+	5.93678e9i	0.364773	+	0.631805i
$708$		0			0
$709$	5.66162e9	−	9.80621e9i	0.596594	−	1.03333i	−0.396726	−	0.917937i	$-0.629854\pi$
$709$	5.66162e9	−	9.80621e9i	0.596594	−	1.03333i	0.993320	−	0.115393i	$-0.0368129\pi$
$710$	−3.57087e8			−0.0374429
$711$		0			0
$712$	7.00052e9			0.726859
$713$	3.87737e9	−	6.71581e9i	0.400612	−	0.693881i
$714$		0			0
$715$	−9.74202e8	−	1.68737e9i	−0.0996731	−	0.172639i
$716$	−4.71746e9	−	8.17088e9i	−0.480300	−	0.831905i
$717$		0			0
$718$	−2.03526e9	+	3.52518e9i	−0.205203	+	0.355423i
$719$	−1.02156e10			−1.02497			−0.512487	−	0.858695i	$-0.671276\pi$
$719$	−1.02156e10			−1.02497			−0.512487	+	0.858695i	$0.671276\pi$
$720$		0			0
$721$	−3.03521e9			−0.301589
$722$	−1.49231e9	+	2.58475e9i	−0.147563	+	0.255587i
$723$		0			0
$724$	2.23368e8	+	3.86885e8i	0.0218744	+	0.0378876i
$725$	−2.52727e9	−	4.37737e9i	−0.246303	−	0.426609i
$726$		0			0
$727$	9.76776e9	−	1.69183e10i	0.942811	−	1.63300i	0.182734	−	0.983162i	$-0.441505\pi$
$727$	9.76776e9	−	1.69183e10i	0.942811	−	1.63300i	0.760076	−	0.649834i	$-0.225161\pi$
$728$	−6.63613e9			−0.637463
$729$		0			0
$730$	1.98036e9			0.188414
$731$	1.54421e10	−	2.67465e10i	1.46216	−	2.53254i
$732$		0			0
$733$	6.63725e9	+	1.14960e10i	0.622478	+	1.07816i	0.989023	+	0.147763i	$0.0472073\pi$
$733$	6.63725e9	+	1.14960e10i	0.622478	+	1.07816i	−0.366545	+	0.930400i	$0.619459\pi$
$734$	−1.63762e9	−	2.83644e9i	−0.152854	−	0.264751i
$735$		0			0
$736$	−4.48607e9	+	7.77010e9i	−0.414757	+	0.718380i
$737$	1.45377e9			0.133770
$738$		0			0
$739$	1.69495e9			0.154490			0.0772452	−	0.997012i	$-0.475388\pi$
$739$	1.69495e9			0.154490			0.0772452	+	0.997012i	$0.475388\pi$
$740$	1.13193e9	−	1.96057e9i	0.102686	−	0.177857i
$741$		0			0
$742$	−2.68828e9	−	4.65624e9i	−0.241580	−	0.418429i
$743$	4.67246e9	+	8.09294e9i	0.417912	+	0.723845i	0.995729	−	0.0923211i	$-0.0294286\pi$
$743$	4.67246e9	+	8.09294e9i	0.417912	+	0.723845i	−0.577817	+	0.816166i	$0.696095\pi$
$744$		0			0
$745$	−1.33958e9	+	2.32023e9i	−0.118692	+	0.205581i
$746$	−2.04785e9			−0.180598
$747$		0			0
$748$	−1.27906e10			−1.11747
$749$	−6.56927e9	+	1.13783e10i	−0.571256	+	0.989444i
$750$		0			0
$751$	6.55969e9	+	1.13617e10i	0.565124	+	0.978824i	0.997038	+	0.0769088i	$0.0245050\pi$
$751$	6.55969e9	+	1.13617e10i	0.565124	+	0.978824i	−0.431914	+	0.901915i	$0.642162\pi$
$752$	1.49165e9	+	2.58362e9i	0.127910	+	0.221547i
$753$		0			0
$754$	−1.55182e9	+	2.68784e9i	−0.131839	+	0.228351i
$755$	−3.48770e9			−0.294934
$756$		0			0
$757$	5.72593e9			0.479745			0.239873	−	0.970804i	$-0.422894\pi$
$757$	5.72593e9			0.479745			0.239873	+	0.970804i	$0.422894\pi$
$758$	−3.05181e9	+	5.28589e9i	−0.254516	+	0.440835i
$759$		0			0
$760$	−1.10910e9	−	1.92101e9i	−0.0916478	−	0.158739i
$761$	7.38104e9	+	1.27843e10i	0.607115	+	1.05155i	0.991713	+	0.128470i	$0.0410066\pi$
$761$	7.38104e9	+	1.27843e10i	0.607115	+	1.05155i	−0.384598	+	0.923084i	$0.625660\pi$
$762$		0			0
$763$	−6.35928e9	+	1.10146e10i	−0.518289	+	0.897703i
$764$	8.14847e9			0.661072
$765$		0			0
$766$	−7.47131e9			−0.600615
$767$	9.43628e8	−	1.63441e9i	0.0755122	−	0.130791i
$768$		0			0
$769$	−2.81775e9	−	4.88049e9i	−0.223440	−	0.387010i	0.732410	−	0.680864i	$-0.238395\pi$
$769$	−2.81775e9	−	4.88049e9i	−0.223440	−	0.387010i	−0.955850	+	0.293854i	$0.905062\pi$
$770$	−9.15389e8	−	1.58550e9i	−0.0722584	−	0.125155i
$771$		0			0
$772$	2.01826e9	−	3.49574e9i	0.157877	−	0.273450i
$773$	4.65338e9			0.362360			0.181180	−	0.983450i	$-0.442008\pi$
$773$	4.65338e9			0.362360			0.181180	+	0.983450i	$0.442008\pi$
$774$		0			0
$775$	−1.11174e10			−0.857918
$776$	−1.10704e10	+	1.91745e10i	−0.850445	+	1.47301i
$777$		0			0
$778$	−8.21033e8	−	1.42207e9i	−0.0625075	−	0.108266i
$779$	−6.60889e9	−	1.14469e10i	−0.500896	−	0.867578i
$780$		0			0
$781$	−1.77250e9	+	3.07007e9i	−0.133140	+	0.230605i
$782$	1.43256e10			1.07125
$783$		0			0
$784$	6.77006e8			0.0501749
$785$	−1.35137e9	+	2.34065e9i	−0.0997084	+	0.172700i
$786$		0			0
$787$	6.17937e9	+	1.07030e10i	0.451890	+	0.782696i	0.998503	−	0.0546889i	$-0.0174167\pi$
$787$	6.17937e9	+	1.07030e10i	0.451890	+	0.782696i	−0.546614	+	0.837385i	$0.684083\pi$
$788$	3.68747e9	+	6.38688e9i	0.268464	+	0.464993i
$789$		0			0
$790$	6.34188e8	−	1.09845e9i	0.0457639	−	0.0792654i
$791$	1.60050e9			0.114984
$792$		0			0
$793$	−1.77364e8			−0.0126302
$794$	1.64599e9	−	2.85094e9i	0.116696	−	0.202123i
$795$		0			0
$796$	−7.40765e9	−	1.28304e10i	−0.520577	−	0.901666i
$797$	−8.52222e9	−	1.47609e10i	−0.596278	−	1.03278i	−0.993365	−	0.115003i	$-0.963312\pi$
$797$	−8.52222e9	−	1.47609e10i	−0.596278	−	1.03278i	0.397087	−	0.917781i	$-0.370021\pi$
$798$		0			0
$799$	−2.24742e10	+	3.89264e10i	−1.55873	+	2.69980i
$800$	1.28627e10			0.888210
$801$		0			0
$802$	−1.84301e10			−1.26159
$803$	9.83008e9	−	1.70262e10i	0.669966	−	1.16041i
$804$		0			0
$805$	−1.25073e9	−	2.16632e9i	−0.0845039	−	0.146365i
$806$	3.41320e9	+	5.91184e9i	0.229609	+	0.397695i
$807$		0			0
$808$	−6.99105e9	+	1.21089e10i	−0.466233	+	0.807539i
$809$	1.00497e10			0.667315			0.333658	−	0.942694i	$-0.391717\pi$
$809$	1.00497e10			0.667315			0.333658	+	0.942694i	$0.391717\pi$
$810$		0			0
$811$	5.26821e9			0.346809			0.173404	−	0.984851i	$-0.444523\pi$
$811$	5.26821e9			0.346809			0.173404	+	0.984851i	$0.444523\pi$
$812$	1.77883e9	−	3.08102e9i	0.116597	−	0.201952i
$813$		0			0
$814$	9.21151e9	+	1.59548e10i	0.598612	+	1.03683i
$815$	1.42740e9	+	2.47233e9i	0.0923620	+	0.159976i
$816$		0			0
$817$	9.42873e9	−	1.63310e10i	0.604889	−	1.04770i
$818$	−6.97814e9			−0.445762
$819$		0			0
$820$	−2.73425e9			−0.173177
$821$	−9.11011e9	+	1.57792e10i	−0.574543	+	0.995138i	0.421548	+	0.906806i	$0.361487\pi$
$821$	−9.11011e9	+	1.57792e10i	−0.574543	+	0.995138i	−0.996091	+	0.0883319i	$0.971846\pi$
$822$		0			0
$823$	6.79338e8	+	1.17665e9i	0.0424802	+	0.0735778i	0.886484	−	0.462760i	$-0.153141\pi$
$823$	6.79338e8	+	1.17665e9i	0.0424802	+	0.0735778i	−0.844004	+	0.536338i	$0.819807\pi$
$824$	−3.09536e9	−	5.36131e9i	−0.192737	−	0.333830i
$825$		0			0
$826$	8.86660e8	−	1.53574e9i	0.0547428	−	0.0948173i
$827$	8.90171e9			0.547273			0.273637	−	0.961833i	$-0.411773\pi$
$827$	8.90171e9			0.547273			0.273637	+	0.961833i	$0.411773\pi$
$828$		0			0
$829$	6.14205e9			0.374432			0.187216	−	0.982319i	$-0.440054\pi$
$829$	6.14205e9			0.374432			0.187216	+	0.982319i	$0.440054\pi$
$830$	4.15566e8	−	7.19782e8i	0.0252271	−	0.0436946i
$831$		0			0
$832$	−4.87806e9	−	8.44905e9i	−0.293640	−	0.508600i
$833$	5.10010e9	+	8.83363e9i	0.305718	+	0.529519i
$834$		0			0
$835$	−1.27297e9	+	2.20486e9i	−0.0756689	+	0.131062i
$836$	−7.80975e9			−0.462291
$837$		0			0
$838$	−8.35455e9			−0.490421
$839$	−5.27318e9	+	9.13341e9i	−0.308252	+	0.533908i	−0.977980	−	0.208698i	$-0.933077\pi$
$839$	−5.27318e9	+	9.13341e9i	−0.308252	+	0.533908i	0.669728	+	0.742606i	$0.266411\pi$
$840$		0			0
$841$	6.27911e9	+	1.08757e10i	0.364009	+	0.630482i
$842$	−8.15942e9	−	1.41325e10i	−0.471050	−	0.815882i
$843$		0			0
$844$	−2.28323e9	+	3.95466e9i	−0.130723	+	0.226418i
$845$	−1.78621e9			−0.101844
$846$		0			0
$847$	−3.78578e9			−0.214074
$848$	1.16633e9	−	2.02015e9i	0.0656805	−	0.113762i
$849$		0			0
$850$	−1.02688e10	−	1.77860e10i	−0.573524	−	0.993373i
$851$	1.25860e10	+	2.17996e10i	0.700058	+	1.21254i
$852$		0			0
$853$	−1.70939e9	+	2.96075e9i	−0.0943018	+	0.163336i	−0.909317	−	0.416104i	$-0.863395\pi$
$853$	−1.70939e9	+	2.96075e9i	−0.0943018	+	0.163336i	0.815015	+	0.579440i	$0.196729\pi$
$854$	−1.66656e8			−0.00915628
$855$		0			0
$856$	−2.67978e10			−1.46030
$857$	1.08519e10	−	1.87960e10i	0.588941	−	1.02007i	−0.405431	−	0.914126i	$-0.632879\pi$
$857$	1.08519e10	−	1.87960e10i	0.588941	−	1.02007i	0.994372	−	0.105949i	$-0.0337881\pi$
$858$		0			0
$859$	1.09605e10	+	1.89842e10i	0.590005	+	1.02192i	0.994231	+	0.107260i	$0.0342076\pi$
$859$	1.09605e10	+	1.89842e10i	0.590005	+	1.02192i	−0.404226	+	0.914659i	$0.632459\pi$
$860$	−1.95044e9	−	3.37826e9i	−0.104565	−	0.181112i
$861$		0			0
$862$	6.14494e9	−	1.06434e10i	0.326770	−	0.565982i
$863$	−1.35347e10			−0.716822			−0.358411	−	0.933564i	$-0.616681\pi$
$863$	−1.35347e10			−0.716822			−0.358411	+	0.933564i	$0.616681\pi$
$864$		0			0
$865$	3.49170e9			0.183434
$866$	4.42418e9	−	7.66290e9i	0.231484	−	0.400941i
$867$		0			0
$868$	−3.91249e9	−	6.77663e9i	−0.203064	−	0.351718i
$869$	−6.29596e9	−	1.09049e10i	−0.325456	−	0.563707i
$870$		0			0
$871$	−8.74274e8	+	1.51429e9i	−0.0448316	+	0.0776505i
$872$	−2.59412e10			−1.32490
$873$		0			0
$874$	8.74701e9			0.443169
$875$	−3.69138e9	+	6.39366e9i	−0.186278	+	0.322642i
$876$		0			0
$877$	−1.22598e10	−	2.12347e10i	−0.613743	−	1.06303i	−0.990604	−	0.136764i	$-0.956330\pi$
$877$	−1.22598e10	−	2.12347e10i	−0.613743	−	1.06303i	0.376861	−	0.926270i	$-0.377004\pi$
$878$	7.90076e9	+	1.36845e10i	0.393947	+	0.682336i
$879$		0			0
$880$	3.97148e8	−	6.87881e8i	0.0196455	−	0.0340270i
$881$	−2.13536e10			−1.05210			−0.526048	−	0.850455i	$-0.676327\pi$
$881$	−2.13536e10			−1.05210			−0.526048	+	0.850455i	$0.676327\pi$
$882$		0			0
$883$	−1.64504e9			−0.0804109			−0.0402054	−	0.999191i	$-0.512801\pi$
$883$	−1.64504e9			−0.0804109			−0.0402054	+	0.999191i	$0.512801\pi$
$884$	7.69205e9	−	1.33230e10i	0.374506	−	0.648664i
$885$		0			0
$886$	8.47042e9	+	1.46712e10i	0.409154	+	0.708676i
$887$	−4.82651e9	−	8.35976e9i	−0.232220	−	0.402217i	0.726241	−	0.687440i	$-0.241266\pi$
$887$	−4.82651e9	−	8.35976e9i	−0.232220	−	0.402217i	−0.958461	+	0.285223i	$0.907932\pi$
$888$		0			0
$889$	3.56955e9	−	6.18264e9i	0.170395	−	0.295133i
$890$	−2.32291e9			−0.110450
$891$		0			0
$892$	−9.90949e9			−0.467492
$893$	−1.37224e10	+	2.37679e10i	−0.644837	+	1.11689i
$894$		0			0
$895$	4.41384e9	+	7.64499e9i	0.205795	+	0.356448i
$896$	3.65375e9	+	6.32849e9i	0.169692	+	0.293915i
$897$		0			0
$898$	−9.59163e9	+	1.66132e10i	−0.442003	+	0.765571i
$899$	−1.03192e10			−0.473683
$900$		0			0
$901$	3.51454e10			1.60078
$902$	1.11254e10	−	1.92698e10i	0.504771	−	0.874289i
$903$		0			0
$904$	1.63222e9	+	2.82709e9i	0.0734833	+	0.127277i
$905$	−2.08992e8	−	3.61984e8i	−0.00937257	−	0.0162338i
$906$		0			0
$907$	4.02709e9	−	6.97512e9i	0.179211	−	0.310403i	−0.762399	−	0.647107i	$-0.775979\pi$
$907$	4.02709e9	−	6.97512e9i	0.179211	−	0.310403i	0.941611	+	0.336704i	$0.109312\pi$
$908$	2.39272e8			0.0106070
$909$		0			0
$910$	2.20200e9			0.0968661
$911$	−2.86899e9	+	4.96924e9i	−0.125723	+	0.217759i	−0.922015	−	0.387153i	$-0.873458\pi$
$911$	−2.86899e9	+	4.96924e9i	−0.125723	+	0.217759i	0.796292	+	0.604912i	$0.206792\pi$
$912$		0			0
$913$	−4.12557e9	−	7.14570e9i	−0.179406	−	0.310740i
$914$	−7.12363e9	−	1.23385e10i	−0.308596	−	0.534503i
$915$		0			0
$916$	−9.71291e8	+	1.68233e9i	−0.0417557	+	0.0723229i
$917$	3.11925e10			1.33585
$918$		0			0
$919$	3.67326e10			1.56116			0.780579	−	0.625057i	$-0.214924\pi$
$919$	3.67326e10			1.56116			0.780579	+	0.625057i	$0.214924\pi$
$920$	2.55102e9	−	4.41850e9i	0.108008	−	0.187076i
$921$		0			0
$922$	−7.49167e9	−	1.29759e10i	−0.314789	−	0.545231i
$923$	−2.13191e9	−	3.69257e9i	−0.0892406	−	0.154569i
$924$		0			0
$925$	1.80435e10	−	3.12523e10i	0.749593	−	1.29833i
$926$	1.21755e10			0.503906
$927$		0			0
$928$	1.19392e10			0.490408
$929$	1.75384e9	−	3.03774e9i	0.0717687	−	0.124307i	−0.827908	−	0.560864i	$-0.810469\pi$
$929$	1.75384e9	−	3.03774e9i	0.0717687	−	0.124307i	0.899677	+	0.436557i	$0.143802\pi$
$930$		0			0
$931$	3.11405e9	+	5.39368e9i	0.126474	+	0.219059i
$932$	1.23608e10	+	2.14096e10i	0.500141	+	0.866270i
$933$		0			0
$934$	−5.02644e9	+	8.70605e9i	−0.201858	+	0.349629i
$935$	1.19674e10			0.478805
$936$		0			0
$937$	−1.68124e10			−0.667638			−0.333819	−	0.942637i	$-0.608337\pi$
$937$	−1.68124e10			−0.667638			−0.333819	+	0.942637i	$0.608337\pi$
$938$	−8.21493e8	+	1.42287e9i	−0.0325008	+	0.0562930i
$939$		0			0
$940$	2.83863e9	+	4.91665e9i	0.111471	+	0.193073i
$941$	−1.75727e10	−	3.04368e10i	−0.687503	−	1.19079i	−0.972643	−	0.232304i	$-0.925374\pi$
$941$	−1.75727e10	−	3.04368e10i	−0.687503	−	1.19079i	0.285141	−	0.958486i	$-0.407960\pi$
$942$		0			0
$943$	1.52011e10	−	2.63290e10i	0.590314	−	1.02245i
$944$	7.69368e8			0.0297668
$945$		0			0
$946$	3.17447e10			1.21914
$947$	−1.65295e10	+	2.86300e10i	−0.632463	+	1.09546i	0.354584	+	0.935024i	$0.384623\pi$
$947$	−1.65295e10	+	2.86300e10i	−0.632463	+	1.09546i	−0.987047	+	0.160434i	$0.948711\pi$
$948$		0			0
$949$	1.18233e10	+	2.04785e10i	0.449062	+	0.777798i
$950$	−6.26995e9	−	1.08599e10i	−0.237264	−	0.410953i
$951$		0			0
$952$	2.03800e10	−	3.52993e10i	0.765553	−	1.32598i
$953$	6.24805e9			0.233840			0.116920	−	0.993141i	$-0.462698\pi$
$953$	6.24805e9			0.233840			0.116920	+	0.993141i	$0.462698\pi$
$954$		0			0
$955$	−7.62401e9			−0.283251
$956$	−1.33364e10	+	2.30993e10i	−0.493669	+	0.855060i
$957$		0			0
$958$	−9.27248e9	−	1.60604e10i	−0.340735	−	0.590170i
$959$	−2.95898e9	−	5.12510e9i	−0.108337	−	0.187645i
$960$		0			0
$961$	2.40787e9	−	4.17055e9i	0.0875187	−	0.151587i
$962$	−2.21586e10			−0.802471
$963$		0			0
$964$	−2.10512e9			−0.0756846
$965$	−1.88837e9	+	3.27074e9i	−0.0676457	+	0.117166i
$966$		0			0
$967$	−1.10632e10	−	1.91621e10i	−0.393449	−	0.681474i	0.599453	−	0.800410i	$-0.295385\pi$
$967$	−1.10632e10	−	1.91621e10i	−0.393449	−	0.681474i	−0.992902	+	0.118936i	$0.962052\pi$
$968$	−3.86081e9	−	6.68711e9i	−0.136809	−	0.236960i
$969$		0			0
$970$	3.67337e9	−	6.36246e9i	0.129230	−	0.223833i
$971$	5.45446e9			0.191199			0.0955993	−	0.995420i	$-0.469523\pi$
$971$	5.45446e9			0.191199			0.0955993	+	0.995420i	$0.469523\pi$
$972$		0			0
$973$	−1.07815e10			−0.375217
$974$	−5.21320e7	+	9.02952e7i	−0.00180779	+	0.00313118i
$975$		0			0
$976$	−3.61525e7	−	6.26180e7i	−0.00124470	−	0.00215588i
$977$	2.47687e10	+	4.29006e10i	0.849711	+	1.47174i	0.881466	+	0.472247i	$0.156557\pi$
$977$	2.47687e10	+	4.29006e10i	0.849711	+	1.47174i	−0.0317547	+	0.999496i	$0.510110\pi$
$978$		0			0
$979$	−1.15304e10	+	1.99713e10i	−0.392741	+	0.680248i
$980$	1.28835e9			0.0437263
$981$		0			0
$982$	4.80086e9			0.161781
$983$	4.36021e9	−	7.55211e9i	0.146410	−	0.253589i	−0.783488	−	0.621407i	$-0.786562\pi$
$983$	4.36021e9	−	7.55211e9i	0.146410	−	0.253589i	0.929898	+	0.367817i	$0.119895\pi$
$984$		0			0
$985$	−3.45013e9	−	5.97581e9i	−0.115029	−	0.199237i
$986$	−9.53153e9	−	1.65091e10i	−0.316660	−	0.548471i
$987$		0			0
$988$	4.69665e9	−	8.13484e9i	0.154931	−	0.268349i
$989$	4.33739e10			1.42574
$990$		0			0
$991$	1.64359e10			0.536458			0.268229	−	0.963355i	$-0.413562\pi$
$991$	1.64359e10			0.536458			0.268229	+	0.963355i	$0.413562\pi$
$992$	1.31300e10	−	2.27418e10i	0.427045	−	0.739664i
$993$		0			0
$994$	−2.00320e9	−	3.46965e9i	−0.0646953	−	0.112056i
$995$	6.93088e9	+	1.20046e10i	0.223053	+	0.386339i
$996$		0			0
$997$	1.59359e10	−	2.76017e10i	0.509264	−	0.882070i	−0.490679	−	0.871341i	$-0.663251\pi$
$997$	1.59359e10	−	2.76017e10i	0.509264	−	0.882070i	0.999942	−	0.0107299i	$-0.00341551\pi$
$998$	1.84797e10			0.588487
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.8.c.d.55.2		4
3.2	odd	2		81.8.c.h.55.1		4
9.2	odd	6		27.8.a.b.1.2	✓	2
9.4	even	3	inner	81.8.c.d.28.2		4
9.5	odd	6		81.8.c.h.28.1		4
9.7	even	3		27.8.a.e.1.1	yes	2
36.7	odd	6		432.8.a.q.1.1		2
36.11	even	6		432.8.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.a.b.1.2	✓	2	9.2	odd	6
27.8.a.e.1.1	yes	2	9.7	even	3
81.8.c.d.28.2		4	9.4	even	3	inner
81.8.c.d.55.2		4	1.1	even	1	trivial
81.8.c.h.28.1		4	9.5	odd	6
81.8.c.h.55.1		4	3.2	odd	2
432.8.a.j.1.2		2	36.11	even	6
432.8.a.q.1.1		2	36.7	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.79669 - 6.57607i) q^{2} +(35.1702 + 60.9166i) q^{4} +(-32.9066 - 56.9959i) q^{5} +(369.202 - 639.477i) q^{7} +1506.08 q^{8} -499.745 q^{10} +(-2480.63 + 4296.58i) q^{11} +(-2983.62 - 5167.78i) q^{13} +(-2803.50 - 4855.80i) q^{14} +(1216.32 - 2106.72i) q^{16} +36651.6 q^{17} +22378.9 q^{19} +(2314.67 - 4009.12i) q^{20} +(18836.4 + 32625.6i) q^{22} +(25736.8 + 44577.4i) q^{23} +(36896.8 - 63907.1i) q^{25} -45311.5 q^{26} +51939.7 q^{28} +(34247.9 - 59319.0i) q^{29} +(-75327.4 - 130471. i) q^{31} +(87152.9 + 150953. i) q^{32} +(139155. - 241024. i) q^{34} -48596.8 q^{35} +489027. q^{37} +(84966.0 - 147165. i) q^{38} +(-49559.9 - 85840.2i) q^{40} +(-295318. - 511505. i) q^{41} +(421321. - 729750. i) q^{43} -348978. q^{44} +390859. q^{46} +(-613184. + 1.06207e6i) q^{47} +(139151. + 241016. i) q^{49} +(-280172. - 485272. i) q^{50} +(209869. - 363504. i) q^{52} +958904. q^{53} +326517. q^{55} +(556047. - 963101. i) q^{56} +(-260057. - 450432. i) q^{58} +(158135. + 273897. i) q^{59} +(14861.5 - 25740.8i) q^{61} -1.14398e6 q^{62} +1.63495e6 q^{64} +(-196362. + 340108. i) q^{65} +(-146512. - 253767. i) q^{67} +(1.28905e6 + 2.23270e6i) q^{68} +(-184507. + 319576. i) q^{70} +714537. q^{71} -3.96273e6 q^{73} +(1.85669e6 - 3.21587e6i) q^{74} +(787073. + 1.36325e6i) q^{76} +(1.83171e6 + 3.17262e6i) q^{77} +(-1.26902e6 + 2.19801e6i) q^{79} -160100. q^{80} -4.48492e6 q^{82} +(-831556. + 1.44030e6i) q^{83} +(-1.20608e6 - 2.08899e6i) q^{85} +(-3.19926e6 - 5.54128e6i) q^{86} +(-3.73602e6 + 6.47098e6i) q^{88} +4.64819e6 q^{89} -4.40624e6 q^{91} +(-1.81034e6 + 3.13560e6i) q^{92} +(4.65614e6 + 8.06467e6i) q^{94} +(-736415. - 1.27551e6i) q^{95} +(-7.35048e6 + 1.27314e7i) q^{97} +2.11325e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 9 q^{2} - 77 q^{4} - 180 q^{5} - 700 q^{7} + 3654 q^{8} + 2790 q^{10} - 10890 q^{11} + 5480 q^{13} - 29475 q^{14} + 15967 q^{16} + 32832 q^{17} + 32048 q^{19} - 12195 q^{20} - 60705 q^{22} - 24372 q^{23}+ \cdots + 45559476 q^{98}+O(q^{100}) \) 4 * q - 9 * q^2 - 77 * q^4 - 180 * q^5 - 700 * q^7 + 3654 * q^8 + 2790 * q^10 - 10890 * q^11 + 5480 * q^13 - 29475 * q^14 + 15967 * q^16 + 32832 * q^17 + 32048 * q^19 - 12195 * q^20 - 60705 * q^22 - 24372 * q^23 + 138880 * q^25 - 470520 * q^26 + 527750 * q^28 + 143280 * q^29 + 38708 * q^31 + 439965 * q^32 + 614088 * q^34 + 231300 * q^35 + 911240 * q^37 + 275382 * q^38 - 135765 * q^40 - 731880 * q^41 + 1088840 * q^43 + 1049130 * q^44 + 3298788 * q^46 - 1561500 * q^47 - 967164 * q^49 + 519660 * q^50 + 2106380 * q^52 + 5220936 * q^53 + 2007000 * q^55 + 650475 * q^56 + 720810 * q^58 - 1731960 * q^59 + 620192 * q^61 - 8572302 * q^62 - 2081678 * q^64 + 914400 * q^65 - 346600 * q^67 + 5559624 * q^68 - 3094425 * q^70 - 8484480 * q^71 - 6290380 * q^73 + 4267710 * q^74 + 2510486 * q^76 - 4864500 * q^77 - 10110616 * q^79 - 3411090 * q^80 - 4282380 * q^82 - 644202 * q^83 - 101520 * q^85 - 2313270 * q^86 - 9374715 * q^88 + 12042000 * q^89 - 41744000 * q^91 - 14795802 * q^92 + 3751290 * q^94 - 747180 * q^95 - 4098670 * q^97 + 45559476 * q^98

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